--- title: "Ellipse through (1,6) and (2,3)" subject: "Mathematics I" chapter: 3 key: "semester_1/mathematics_i:ellipse-through-two-points" source: https://www.bctengineer.com/q/semester_1/mathematics_i/ellipse-through-two-points site: BCT Engineering --- # Ellipse through (1,6) and (2,3) Mathematics I · Chapter 3: Plane Analytic Geometry - Asked in: 2069 Asar - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/mathematics_i/ellipse-through-two-points - Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-3 ## Question Find the equation of the ellipse whose centre is the origin and whose axes are the axes of coordinates and which passes through the points (1, 6) and (2, 3). ## Short answer Substitute points into x²/a² + y²/b² = 1, solve a²=5, b²=45. Equation: x²/5 + y²/45 = 1 or 9x² + y² = 45. ## Answer An ellipse with centre at the origin and axes along the coordinate axes has the form x²/a² + y²/b² = 1 Let A = 1/a² and B = 1/b². Then Ax² + By² = 1. #### Use the two points (1, 6): A + 36B = 1 ... (i) (2, 3): 4A + 9B = 1 ... (ii) From (i), A = 1 − 36B. Substituting in (ii): 4 − 144B + 9B = 1 135B = 3, so B = 1/45 A = 1 − 36/45 = 9/45 = 1/5 So a² = 5 and b² = 45. **Required ellipse: x²/5 + y²/45 = 1, i.e. 9x² + y² = 45.** Check: (1, 6): 9 + 36 = 45 ✓. (2, 3): 36 + 9 = 45 ✓. Here b² > a², so the major axis lies along the y-axis (semi-major 3√5, semi-minor √5). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which trick makes the equations linear? A. Square both points B. Differentiate C. Let A = 1/a^2 and B = 1/b^2 so Ax^2 + By^2 = 1 D. Put x = y Answer: C. The points give two linear equations in A and B. (Key point: Set-up) 2. Which equations come from (1,6) and (2,3)? A. A + 6B = 1 and 2A + 3B = 1 B. A + 36B = 1 and 4A + 9B = 1 C. 6A + B = 1 and 3A + 2B = 1 D. A + 36B = 0 and 4A + 9B = 0 Answer: B. Substitute x^2 and y^2 values of each point. (Key point: Equations) 3. What are a^2 and b^2? A. a^2 = 5, b^2 = 45 B. a^2 = 45, b^2 = 5 C. a^2 = 1/5, b^2 = 1/45 D. a^2 = 9, b^2 = 1 Answer: A. B = 1/45, A = 1/5. (Key point: Solution) 4. Since b^2 > a^2, where is the major axis? A. Along the x-axis B. Along y = x C. It is a circle D. Along the y-axis Answer: D. Larger denominator is under y^2: 9x^2 + y^2 = 45. (Key point: Which axis is major) ## Related in Mathematics I - [Centre, axes of ellipse 2x²+3y²−4x−12y+13=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aellipse-2x2-3y2-centre-axes) — Mathematics I, chapter 3, asked 7× - [Centre, axes, directrices of 3x²+8xy−3y²−40x−20y+50=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-3x2-8xy-3y2-directrices) — Mathematics I, chapter 3, asked 7× - [Centre, axes of 9x²+4xy+6y²−22x−16y+9=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-9x2-4xy-6y2-centre-axes) — Mathematics I, chapter 3, asked 7× - [Derive standard equation of an ellipse](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aellipse-standard-equation-derive) — Mathematics I, chapter 3, asked 6× - [Pedal equation of an ellipse](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Apedal-equation-ellipse) — Mathematics I, chapter 1 - [Equation and length of axes of x²−4xy−2y²+10x+4y=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-x2-4xy-2y2-equation-axes) — Mathematics I, chapter 3 ## Related in other subjects - [Checking whether root locus passes through point P](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-point-p-dominant) — Control Systems, chapter 5 - [Parametric circle and ellipse with points](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Acircle-ellipse-parametric) — Computer Aided Design, chapter 3 - [Sphere through circle of two spheres, centre on a plane](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Asphere-through-circle-two-spheres) — Mathematics, chapter 3 - [Plane through three points and distance](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Aplane-surface-three-points) — Computer Aided Design, chapter 4